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Theory and Applications of Categories, Vol. 33, No. 7, 2018, pp. 148–192.

REGULAR PATTERNS, SUBSTITUDES,FEYNMAN CATEGORIES AND OPERADS

MICHAEL BATANIN, JOACHIM KOCK, AND MARK WEBER

Abstract. We show that the regular patterns of Getzler (2009) form a 2-categorybiequivalent to the 2-category of substitudes of Day and Street (2003), and that theFeynman categories of Kaufmann and Ward (2013) form a 2-category biequivalent tothe 2-category of coloured operads (with invertible 2-cells). These biequivalences induceequivalences between the corresponding categories of algebras. There are three mainingredients in establishing these biequivalences. The first is a strictification theorem(exploiting Power’s General Coherence Result) which allows to reduce to the case wherethe structure maps are identity-on-objects functors and strict monoidal. Second, wesubsume the Getzler and Kaufmann–Ward hereditary axioms into the notion of Guitartexactness, a general condition ensuring compatibility between certain left Kan exten-sions and a given monad, in this case the free-symmetric-monoidal-category monad.Finally we set up a biadjunction between substitudes and what we call pinned symmet-ric monoidal categories, from which the results follow as a consequence of the fact thatthe hereditary map is precisely the counit of this biadjunction.

0. Introduction and overview of results

A proliferation of operad-related structures have seen the light in the past decades, suchas modular and cyclic operads, properads, and props. Work of many people has soughtto develop categorical formalisms covering all these notions on a common footing, and inparticular to describe adjunctions induced by the passage from one type of structure toanother as a restriction/Kan extension pair [3, 4, 6, 8, 11, 14, 17, 18, 19, 31, 38]. For the lineof development of the present work, the work of Costello [11] was especially inspirational:in order to construct the modular envelope of a cyclic operad, he presented these notionsas symmetric monoidal functors out of certain symmetric monoidal categories of treesand graphs, and arrived at the modular envelope as a left Kan extension correspondingto the inclusion of one symmetric monoidal category into the other. Unfortunately it isnot clear from this construction that the resulting functor is even symmetric monoidal.The problem was addressed by Getzler [19] by identifying a condition needed for theconstruction to work: he introduced the notion of a ‘regular pattern’ (cf. 0.1 below),which includes a condition formulated in terms of Day convolution, and which guaranteesthat constructions like Costello’s will work. However, his condition is not always easy

Received by the editors 2017-05-25 and, in final form, 2018-02-11.Transmitted by Ross Street. Published on 2018-02-19.2010 Mathematics Subject Classification: 18D10, 18D50.Key words and phrases: operads, symmetric monoidal categories.c© Michael Batanin, Joachim Kock, and Mark Weber, 2018. Permission to copy for private use

granted.

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REGULAR PATTERNS, SUBSTITUDES, FEYNMAN CATEGORIES, OPERADS 149

to verify in practice. Meanwhile, Markl [31], and later Borisov and Manin [8], studiedgeneral notions of graph categories, designed with generalised notions of operad in mind,and isolated in particular a certain hereditary condition, which has also been studied byMellies and Tabareau [32] within a different formalism (cf. 3.2 below). This conditionfound a comma-category formulation in the recent work of Kaufmann and Ward [25],being the essential axiom in their notion of ‘Feynman category’, cf. 0.2 below.

Kaufmann and Ward notice that the hereditary axiom is closely related to the Day-convolution Kan extension property of Getzler, and provide an easy-to-check conditionunder which the envelope construction (and other constructions given by left Kan exten-sions) work. Their work is the starting point for our investigations.

Another common generalisation of operads and symmetric monoidal categories arethe substitudes of Day and Street [14, 15] (in fact considered briefly already by Baez andDolan [1] under the name C-operad). Their interest came from the study of a nonstandardconvolution construction introduced by Bakalov, D’Andrea, and Kac [2]. Substitudes canbe also understood as monads in the bicategory of generalised species, introduced byFiore, Gambino, Hyland and Winskel [17] in 2008.

In the present paper we prove that regular patterns are essentially the same thingas substitudes, and that Feynman categories are essentially the same thing as (coloured)operads. More precisely, we establish biequivalences of 2-categories—this is the bestsameness one can hope for, since the involved structures are categorical and hence form2-categories. For all four notions, a key aspect is their algebras. We show furthermorethat under the biequivalences established, the notions of algebras agree. More precisely, ifa regular pattern and a substitude correspond to each other under the biequivalence, thentheir categories of algebras are equivalent. Similarly of course with Feynman categoriesand operads.

In a broader perspective, our results can be seen as part of a dictionary between twoapproaches to operad-like structures and their algebras, namely the symmetric-monoidal-category approach and the operadic/multicategorical approach. This dictionary goes backto the origins of operad theory, cf. Chapter 2 of Boardman–Vogt [7]. In fact to establishthe results we exploit a third approach, namely that of 2-monads, which goes back toKelly’s paper on clubs [26]. This more abstract approach allows us to pinpoint someessential mechanisms in both approaches. In particular we subsume the Getzler andKaufmann–Ward hereditary axioms into the 2-categorical notion of Guitart exactness, ageneral condition ensuring compatibility between certain left Kan extensions and a given2-monad, in this case the free-symmetric-monoidal-category monad. Since the notion ofGuitart exactness has recently proved very useful in operad theory and abstract homotopytheory [4, 20, 24, 30, 38], this interpretation of the axioms of Getzler and Kaufmann–Wardis of independent interest, and we elaborate on it in some detail.

The equivalence between regular patterns and substitudes does not seem to have beenforeseen by anybody. The equivalence between Feynman categories and operads maycome as a surprise, as Kaufmann and Ward in fact introduced Feynman categories withthe intention of providing an ‘improvement’ over the theory of operads. Part of the

150 MICHAEL BATANIN, JOACHIM KOCK, AND MARK WEBER

structure of Feynman category is an explicit groupoid, which one might think of as a‘groupoid of colours’, in contrast to the set of colours of an operad. Rather unexpectedly,this groupoid is now revealed to be available already in the usual notion of operad, namelyas the groupoid of invertible unary operations. Since the notion of operad is undoubtedlyfundamental, the equivalence we establish attests to the importance also of the notion ofFeynman category, now to be regarded as a useful alternative viewpoint on operads.

In the present paper, for the sake of focusing on the principal ideas, we work only overthe category of sets. For the enriched setting, we refer to Caviglia [10], who independentlyhas established an enriched version of the equivalence between Feynman categories andoperads.

We proceed to state our main result, and sketch the ingredients that go into its proof.

For C a category, we denote by SC the free symmetric monoidal category on C.

0.1. Definition of regular pattern. (Getzler [19]) A regular pattern is a symmetricstrong monoidal functor τ : SC ÑM such that

(1) τ is essentially surjective

(2) the induced functor of presheaves τ� : xM Ñ xSC is strong monoidal for the Dayconvolution tensor product.

0.2. Definition of Feynman category. (Kaufmann–Ward [25]) A Feynman categoryis a symmetric strong monoidal functor τ : SC ÑM such that

(1) C is a groupoid(2) τ induces an equivalence of groupoids SC �ÑMiso

(3) τ induces an equivalence of groupoids SpMÓCqiso�Ñ pMÓMqiso.

0.3. The hereditary condition. Getzler’s definition is staged in the enriched setting.Kaufmann and Ward also give an enriched version called weak Feynman category ([25],Definition 4.2 and Remark 4.3) which over Set reads as follows:

τ : SC Ñ M is an essentially surjective symmetric strong monoidal functor, and thefollowing important hereditary condition holds (formulated in more detail in 3.2): For anyx1, . . . , xm, y1, . . . , yn P C, the natural map given by tensoring

¸α:mÑn

¹jPn

Mpâ

iPα�1pjq

τxi, τyjq ÝÑMpâiPm

τxi,âjPn

τyjq

is a bijection. (They recognise that this weak notion is ‘close’ to Getzler’s notion of regularpattern but do not prove that it is actually equivalent. In fact this condition does notreally play a role in the developments in [25].)

The hereditary condition is natural from a combinatorial viewpoint where it says thatevery morphism splits into a tensor product of ‘connected’ morphisms. We shall see (5.13)that in the essentially surjective case it is exactly the condition that the counit for thesubstitude Hermida adjunction is fully faithful.


0.4. Operads and substitudes. By operad we mean coloured symmetric operad inSet. We refer to the colours as objects. The notion of substitude was introduced byDay and Street [14], as a general framework for substitution in the enriched setting. Oursubstitudes are their symmetric substitudes, cf. also [5], whose appendix constitutes aconcise reference for the basic theory of substitudes. A quick definition is this (cf. [15,6.3]): a substitude is an operad equipped with an identity-on-objects operad morphismfrom a category (regarded as an operad with only unary operations).

We can now state the main theorem:

Theorem. (Cf. Theorem 5.14 and Theorem 5.16.) There is a biequivalence between the 2-category of substitudes and the 2-category of regular patterns. It restricts to a biequivalencebetween the 2-category of operads (with invertible 2-cells) and the 2-category of Feynmancategories (with invertible 2-cells).

The biequivalence means that when going back and forth, not an isomorphic object isobtained, but only an equivalent one. This is a question of strictification: one ingredientin the proof is to show that every regular pattern is equivalent to a strict one, and avariant of the main theorem can be stated as a 1-equivalence between these strict regularpatterns and substitudes. It should be observed that equivalent regular patterns haveequivalent algebras (5.19).

We briefly run through the main ingredients of the proof, and outline the contents ofthe paper.

In Section 1, we show that regular patterns and Feynman categories can be strictified.Both notions concern a symmetric strong monoidal functor τ : SC ÑM where SC is thefree symmetric monoidal category on a category C, and in particular is strict. The mainresult is this:

Proposition. (Cf. Proposition 1.6.) Every essentially surjective symmetric strong mon-oidal functor SC ÑM , is equivalent to one SC ÑM 1, for which M 1 is a symmetric strictmonoidal category, and SC ÑM 1 is strict monoidal and identity-on-objects.

This is a consequence of Power’s coherence result [33], recalled the appendix. Since thenotions of regular pattern and Feynman category are invariant under monoidal equiva-lence, we may as well work with the strict case, which will facilitate the arguments greatly,and highlight the essential features of the notions, over the subtleties of having coherenceisomorphisms everywhere.

The next step, which makes up Section 2, is to put Getzler’s condition (2) into thecontext of Guitart exactness.

0.5. Guitart exactness. Guitart [21] introduced the notion of exact square: they arethose squares that pasted on top of a pointwise left Kan extension again gives a pointwiseleft Kan extension. A morphism of T -algebras for a monad T is exact when the algebramorphism coherence square is exact. We shall need this notion only in the case where


T is the free symmetric monoidal category monad on Cat: it thus concerns symmetricmonoidal functors.

Theorem. (Cf. Theorem 2.11.) The following are equivalent for a symmetric colaxmonoidal functor τ : S ÑM .

(1) τ is Guitart exact(2) left Kan extension of Yoneda along τ is strong monoidal(3) left Kan extension of any strong monoidal functor along τ is again strong monoidal

(4) τ� : xM Ñ pS is strong monoidal(5) a certain category of factorisations is connected (cf. Lemmas 2.4 and 3.9)

In the special case of interest to us we thus have

Corollary. For a symmetric strong monoidal functor τ : SC Ñ M , axiom (2) of beinga regular pattern is equivalent to being exact.

In Section 3 we analyse the hereditary condition, also shown to be equivalent to aspecial case of Guitart exactness:

Proposition. (Cf. Proposition 3.3.) An essentially surjective symmetric strong monoidalfunctor τ : SC ÑM is exact if and only if it satisfies the hereditary condition.

Corollary. A regular pattern is a symmetric strong monoidal functor τ : SC Ñ Mwhich is essentially surjective and satisfies the hereditary condition.

Axiom (3) of the notion of Feynman category of Kaufmann and Ward [25], the equiva-lence of comma categories SpMÓCqiso

�Ñ pMÓMqiso, is of a slightly different flavour to theother related conditions (and in particular, does not seem to carry over to the enrichedcontext). While it is implicit in [25] that this condition is essentially equivalent to thehereditary condition, the relationship is actually involved enough to warrant a detailedproof, which makes up our Section 4.

The outcome is the following result, essentially proved by Kaufmann and Ward [25].

Corollary. A Feynman category is a special case of a regular pattern, namely such thatC is a groupoid and SC ÑMiso is an equivalence.

With these two corollaries in place, we can finally establish the promised biequivalencesin Section 5. We achieve this by setting up pinned variations of the symmetric Hermidaadjunction [22] between symmetric monoidal categories and operads:

0.6. Pinned symmetric monoidal categories and pinned operads. A pinnedsymmetric monoidal category is a symmetric monoidal category M equipped with a sym-metric strong monoidal functor SC Ñ M (where SC is the free symmetric monoidalcategory on some category C). Hence regular patterns and Feynman categories are ex-amples of pinned symmetric monoidal categories. Similarly, a pinned operad is defined tobe an operad equipped with a functor from a category, viewed as an operad with only


unary operations. Substitudes are thus pinned operads for which the structure map isidentity-on-objects. The latter condition exhibits substitudes as a coreflective subcategoryof pinned operads.

0.7. The substitude Hermida adjunction. Our main result will follow readily fromthe following variation on the Hermida adjunction—actually a biadjunction, which goesbetween pinned symmetric monoidal categories and substitudes via pinned operads:

pSMC //K pOpdoo

//K Subst.oo

(1)

The right adjoint takes a pinned symmetric monoidal category SC Ñ M to the sub-stitude

C Ñ EndpMq|C,

the endomorphism operad on M , base-changed to C. It is a important feature of substi-tudes (not enjoyed by operads) that they can be base-changed along functors.

The left adjoint in (1) takes a substitude C Ñ P to the pinned symmetric monoidalcategory SC Ñ FP , where FP is the free symmetric monoidal category on P as in theordinary Hermida adjunction: the objects of FP are finite sequences of objects in P , andits arrows from sequence x1, . . . , xm to sequence y1, . . . , yn are given by

FP px,yq :�¸

α:mÑn

¹jPn

P ppxiqiPα�1pjq, yjq.

The left adjoint is now shown to be fully faithful (5.9). This important feature is notshared by the original Hermida adjunction. Our key result characterises the image of theleft adjoint by determining where the counit is invertible:

Proposition. (Cf. Proposition 5.12.) The counit ετ is an equivalence if and only if τ isessentially surjective and the hereditary condition holds.

Corollary. The essential image of the left adjoint is the 2-category of regular patterns.

In particular, this establishes the first part of the Main Theorem:

Theorem. (Cf. Theorem 5.14.) The left adjoint induces a biequivalence between the 2-category of substitudes and the 2-category of regular patterns.

To an operad P one can assign a substitude by taking the canonical groupoid pinningP iso

1 Ñ P . This is not functorial in all 2-cells, only in invertible ones; it is the object partof a fully faithful 2-functor pOpdq2-iso Ñ Subst. We characterise its regular patterns inthe image of this 2-functor: they are precisely the Feynman categories. This establishesthe second part of the main theorem:


Theorem. (Cf. Theorem 5.16.) The previous biequivalence induces a biequivalence be-tween the 2-category of operads (with invertible 2-cells) and the 2-category of Feynmancategories (with invertible 2-cells).

Acknowledgments. The authors thank Ezra Getzler, Ralph Kaufmann, and RichardGarner for fruitful conversations, and thank the anonymous referee for many pertinentremarks and suggestions that led to some simplifications. The bulk of this work wascarried out while J.K. was visiting Macquarie University in February–March 2015, spon-sored by Australian Research Council Discovery Grant DP130101969. M.B. acknowl-edges the financial support of Scott Russell Johnson Memorial Foundation, J.K. was sup-ported by grant number MTM2013-42293-P of Spain, and M.W. by grant number GA CRP201/12/G028 from the Czech Science Foundation. Both M.B. and M.W. acknowledgethe support of the Australian Research Council grant No. DP130101172.

1. Strictification of regular patterns

1.1. The free symmetric monoidal category. The free symmetric monoidal cat-egory SC on a category C has the following explicit description. The objects of SC arethe finite sequences of objects of C. A morphism is of the form

pρ, pfiqiPnq : pxiqiPn ÝÑ pyiqiPn

where ρ P Σn is a permutation, and for i P n � t1, ..., nu, fi : xi Ñ yρi. Intuitively such amorphism is a permutation labelled by arrows of C, as in

x1 x2 x3 x4

y1 y2 y3 y4.

((f1

��f2 �� f3

zzf4

For further details, see the Appendix, where it is explained and exploited that S underliesa 2-monad on Cat. It will be important that SC is actually a symmetric strict monoidalcategory.

1.2. Gabriel factorisation. Given a functor F : C Ñ D, its factorisation into anidentity-on-objects followed by a fully faithful functor is referred to as the Gabriel fac-torisation of F :

CF //

i.o.

D

D1f.f.

>>


1.3. Proposition. Let F : S ÑM be a symmetric strong monoidal functor, and assumethat S is a symmetric strict monoidal category. Then for the Gabriel factorisation

SF //

G

M

M 1H

==

there is a canonical symmetric strict monoidal structure on M 1 for which G is a symmetricstrict monoidal functor, and H is canonically symmetric strong monoidal.

The proof, relegated to the Appendix, exploits the general coherence result of Power [33].A direct proof is somewhat subtle because the monoidal structure on M 1 is constructedas a mix of the monoidal structures on S and on M , and it is rather cumbersome to checkthat the trivial associator defined on M 1 is actually natural. Instead following Power’sapproach gives an elegant abstract proof, which exploits the following easily checked facts:(1) the Gabriel factorisation has a 2-dimensional aspect where isomorphisms can alwaysbe shifted right in the factorisation; and (2) S preserves this factorisation.

1.4. Pinned symmetric monoidal categories. The following terminology will bejustified in Section 5, as part of further pinned notions. A pinned symmetric monoidalcategory is a symmetric monoidal category M equipped with a symmetric strong monoidalfunctor τ : SC ÑM (where C is some category). Pinned symmetric monoidal categoriesare the objects of a 2-category pSMC . A morphism pC1, τ1,M1q Ñ pC2, τ2,M2q is atriple pF,G, ωq consisting of a functor F : C1 Ñ C2, a symmetric strong monoidal functorG : M1 ÑM2, and an invertible monoidal natural transformation ω : τ2�SF � G�τ1. A 2-cell pF,G, ωq Ñ pF 1, G1, ω1q is a pair pα, βq, where α : F Ñ F 1 is a natural transformationand β : G Ñ G1 is a monoidal natural transformation, such that ω pasted with β equalsSα pasted with ω1.

A pinned symmetric monoidal category τ : SC Ñ M is called strict when M is asymmetric strict monoidal category and τ is a symmetric strict monoidal functor. Amorphism pF,G, ωq : pC1, τ1,M1q Ñ pC2, τ2,M2q is strict when G is a symmetric strictmonoidal functor and ω is the identity. The locally full sub-2-category spanned by thestrict objects and strict morphisms is denoted pSMCs.

1.5. Regular patterns. (Getzler [19]) A regular pattern is a symmetric strong monoidalfunctor τ : SC ÑM such that

(1) τ is essentially surjective

(2) the induced functor of presheaves τ� : xM Ñ xSC is strong monoidal for the Dayconvolution tensor product.

Regular patterns form a 2-category RPat, namely the full sub-2-category of pSMCspanned by the regular patterns.

A regular pattern (resp. a morphism of regular patterns) is called strict when it isstrict as a pinned symmetric monoidal category (resp. a morphism of pinned symmetric


monoidal categories). These form thus a full sub-2-category RPats � pSMCs and alocally full sub-2-category RPats � RPat.

1.6. Proposition. The inclusion 2-functor RPats � RPat is a biequivalence.

Proof. If τ : SC Ñ M is a regular pattern, in its Gabriel factorisation (as in Proposi-tion 1.3)

SC τ //

σ!!

M

M 1φ

==

σ is identity-on-objects, and φ is a strong monoidal equivalence (say with pseudo-inverse

ψ and 2-cell ω : ψφ � idM 1). It follows that σ� : xM Ñ xSC is strong monoidal: in anycase it is lax monoidal, and since both τ� and φ� are strong monoidal, also σ� is strongmonoidal. In other words, σ is a strict regular pattern. The triple pidC , φ, idq : σ Ñ τ isan equivalence in RPat with pseudo-inverse pidC , ψ, ωσq. This shows that τ is equivalentto a strict regular pattern, so the inclusion 2-functor is essentially surjective on objects.

The inclusion 2-functor is locally fully faithful by construction, so it remains to see itis locally essentially surjective, i.e. essentially surjective on morphisms. We need to show,given strict regular patterns σ and σ1, that any morphism pF,G, ωq : σ Ñ σ1 is equivalentto a strict one pF,Gstrict, ωstrict� idq. Since σ is bijective on objects, there is a unique wayto define the strict Gstrict on objects so that ωstrict becomes the identity 2-cell. It will beequivalent to G by means of the old ω, which also ensures the functoriality of Gstrict. Itis symmetric strict monoidal since SF and σ1 are. So the inclusion 2-functor is locallyessentially surjective, and hence altogether a biequivalence.

1.7. Algebras. Let W be a symmetric monoidal category. An algebra for a regularpattern τ : SC Ñ M in W , is a symmetric strong monoidal functor M Ñ W . Withmorphisms of algebras given by monoidal natural transformations, there is a categoryAlgτ pW q of algebras of pC, τ,Mq in W . A morphism of regular patterns pC, τ,Mq ÑpC 1, τ 1,M 1q induces a functor Algτ 1pW q Ñ Algτ pW q by precomposition. The followingproposition is now clear.

1.8. Proposition. Equivalent regular patterns have equivalent categories of algebras.

Together with Proposition 1.6, this justifies emphasising strict regular patterns, as weshall often do. This facilitates extracting equivalent characterisations of condition (2)—Guitart exactness and the hereditary condition, in turn exploited in the final comparisonwith substitudes.

2. Guitart exactness

In this section and the next we show how the main axioms in the definitions of regularpattern and Feynman category can be subsumed in the theory of Guitart exactness.


An important aspect of Guitart exactness is to serve as a criterion for pointwise leftKan extensions to be compatible with algebraic structures. This direction of the theoryis developed rather systematically in [38] in the abstract setting of a 2-monad on a 2-category with comma objects. The interesting case for the present purposes is the case ofthe free-symmetric-monoidal-category monad on Cat, and the issue is then under whatcircumstances left Kan extensions are symmetric monoidal functors.

We write pA :� rAop,Sets for the category of presheaves, and yA : A Ñ pA for theYoneda embedding.

2.1. Exact squares. A 2-cell in Cat of the form

P B

CA

q//

g��//

f

��

p φ +3 (2)

(called a lax square) is exact in the sense of Guitart [21] when for any natural transfor-mation ψ which exhibits l as a pointwise left Kan extension of h along f , the composite

P B

A C

V

f//

l��h

ψ +3

q//

g��

p��

φ +3

exhibits lg as a pointwise left Kan extension of hp along q.Suppose that in this situation A is locally small and f is admissible in the sense [34, 35]

that Cpfa, cq is small for all a P A and c P C. One thus has the functor Cpf, 1q : C Ñ pAgiven on objects by c ÞÑ Cpfp�q, cq, and the effect on arrows of the functor f can beorganised into a natural transformation

A C

pA

f//

Cpf,1q��yA

χf +3

which exhibits Cpf, 1q as a pointwise left Kan extension of yA along f (see e.g. [35]Example 3.3).

2.2. Lemma. (Cf. Guitart [21].) A lax square (2) in which A and P are small and f is


admissible is exact if and only if the composite

P B

A C

pA

f//

Cpf,1q��yA

χf +3

q//

g��

p��

φ +3

(3)

exhibits Cpf, 1q � g as a pointwise left Kan extension of yA � p along q.

When f � yA, the 2-cell χf is the identity, and we get the following.

2.3. Corollary. If P and A are small and

P B

pAA

q//

g��

//yA

��

p φ +3

exhibits g as a pointwise left Kan extension of yA � p along q, then φ is exact.

Exact squares can be recognised in elementary terms in the following way. First givena P A, b P B and γ : faÑ gb we denote by Factφpa, γ, bq the following category. Its objectsare triples pα, x, βq where x P P , α : a Ñ px and β : qx Ñ b, such that gpβqφxfpαq � γ.Informally, such an object is a ‘factorisation of γ through φ’. A morphism pα1, x1, β1q Ñpα2, x2, β2q of such is an arrow δ : x1 Ñ x2 such that ppδqα1 � α2 and β1 � β2qpδq.Identities and compositions are inherited from P .

2.4. Lemma. [21] A lax square (2) in Cat is exact if and only if for all a P A, b P B,and γ : faÑ gb, the category Factφpa, γ, bq defined above is connected.

2.5. Exact monoidal functors. In the usual nullary-binary way of writing tensorproducts in monoidal categories, a symmetric colax monoidal functor f : A Ñ B hascoherence morphisms of the form

f 0 : fI ÝÑ I fX,Y : fpX b Y q ÝÑ fX b fY

(in which I denotes the unit of either A or B) which are required to satisfy axioms thatexpress compatibility with the coherences which define the symmetric monoidal structureson A and B. Equivalently one can regard a symmetric colax monoidal structure on f ascomprising coherence morphisms

fX1,...,Xn : fpX1 b � � � bXnq ÝÑ fpX1q b � � � b fpXnq


for each sequence pX1, ..., Xnq of objects of A, whose naturality is expressed by the factthat they are the components of a natural transformation

SA SB

B.A

Spfq//

Â��//

f

��

Â f +3

We say that f is exact when this square is an exact square in the sense discussed above.In terms of the 2-monad S, pf, fq is a colax morphism of pseudo algebras, and in [38], thetheory of exact colax morphisms of algebras is developed at the general level of a 2-monadon a 2-category with comma objects.

The following lemma is key to the interest in exactness in the present context. Theresult is a special case of Theorem 2.4.4 of [38]. A similar result is obtained in the contextof proarrow equipments in [32] and in a double categorical setting in [29].

2.6. Lemma. [38] Let f : AÑ B be an exact symmetric colax monoidal functor. Then forany lax symmetric monoidal functor g : AÑ C (with C assumed algebraically cocomplete),the pointwise left Kan extension lanf g

Af

//

g��

ñ

B

lanf g��

C

is again naturally lax symmetric monoidal. Furthermore, if g is strong, then so is lanf g.

The condition that C is algebraically cocomplete (with respect to f) means first of allthat it has enough colimits for the left Kan extension in question to exist, and second,that these colimits are preserved by the tensor product in each variable. More formally,whenever ψ exhibits h as a pointwise left Kan extension of g along f as on the left, thenthe composite on the right

A B

C

f//

h��g

ψ +3

SA SB

SC

C

Sf//

Sh��Sg

Â��

Sψ +3

exhibitsÂ

�Sh as a pointwise left Kan extension ofÂ

�Sg along Sf .


2.7. Day convolution tensor product. It is well known that the symmetric monoid-al structure on A (assumed to be small) extends essentially uniquely to one on pA, for which

the tensor product is cocontinuous in each variable. This tensor product � : S pA Ñ pA iscalled Day convolution [13]. It is folklore that the Day convolution tensor product canalso be characterised as a pointwise left Kan extension as in the following result, which isnothing more than a translation of the universal property of convolution as expressed byIm and Kelly [23], in these terms.

2.8. Proposition. [23] For A a small symmetric monoidal category, the Day convolution

tensor product � on pA can be characterised as the pointwise left Kan extension of yA �Â

along SyA,

SA S pA

pA.A

SyA //

�

��

//yA

��

Â yA +3

Furthermore, this square (invertible since SyA is fully faithful) constitutes the coherencedata making yA a symmetric strong monoidal functor. Finally, the following universalproperty holds (usually taken as the defining property of the Day convolution tensor prod-uct): For any cocomplete symmetric monoidal category X, composition with yA givesequivalences of categories

CoctsSMCcp pA,Xq � SMCcpA,Xq CoctsSMCp pA,Xq � SMCpA,Xq.

Here SMC is the 2-category of symmetric monoidal categories with symmetric strongmonoidal functors, while SMCc has also symmetric colax monoidal functors. The pre-fixes Cocts indicate the full subcategories spanned by cocomplete symmetric monoidalcategories whose tensor product preserves colimits in both variables.

Proof. For any category C, we denote by MC the free (strict) monoidal category onC. Explicitly MC is the subcategory of SC containing all the objects, but just themorphisms whose underlying permutation is an identity. The inclusions iC : MC Ñ SCare the components of a 2-natural transformation i : M Ñ S which by the results of [36],conforms to the hypotheses of Proposition 4.6.2 of [38]. Thus for any functor f : C Ñ D,the corresponding naturality square of i on the left

MC MD

SDSC

Mf//

iD��

//

Sf

��

iC SA S pA

pAA

MA M pASyA //

�

��

//yA

��

Â

iA��

MyA //

ipA��

yA +3

is exact, and so the composite square on the right exhibits � � i pA as a pointwise leftKan extension, and this functor has the same object map as �. Computing the left Kan


extension on the left in the previous display as a coend in the usual way, one recovers theusual formula for the Day tensor product. Thus the result follows from [23].

With Corollary 2.3, we arrive at the following.

2.9. Corollary. yA : pA,bq Ñ p pA,�q is exact.

2.10. Generalities. For any functor f : AÑ B between small categories, we have the2-cells

Af

//

yA

��

χf

ñ

ByB //

Bpf,1q

��

pBιf

ñ

f�

��pA

AyA //

f

��

lf

ñ

pAf!��

B yB// pB

exhibiting Bpf, 1q as the pointwise left Kan extension of yA along f , and f� (restrictionalong f) as the pointwise left Kan extension of Bpf, 1q along yB. Finally, lf exhibits f!

(the left adjoint to f�) as the pointwise left Kan extension of yB � f along yA. Note thatlf is an exact square by Corollary 2.3, and that both ιf and lf are invertible, since yB andyA are fully faithful.

Returning to our situation of a symmetric colax monoidal functor between small sym-metric monoidal categories pf, fq : A Ñ B, by Lemma 2.6 f! gets a symmetric colaxmonoidal structure from that of yB � f , since yA is exact by Proposition 2.8. The colaxcoherence datum f ! (which we don’t make explicit here, and which is invertible if andonly if f is) induces, by taking mates via f! % f�, the coherence 2-cell f

�making f� a

lax monoidal functor. Moreover Bpf, 1q gets a unique monoidal structure making ιf aninvertible monoidal natural transformation. In the context just described we have the fol-lowing alternative characterisations of exactness of the symmetric colax monoidal functorpf, fq.

2.11. Theorem. The following statements are equivalent for a colax symmetric monoidalfunctor f : AÑ B (assuming A small and f admissible).

(1) f is exact.(2) For any algebraically cocomplete symmetric monoidal category X and any symmetric

strong monoidal functor g : A Ñ X, the pointwise left Kan extension of g along fis symmetric strong monoidal.

(3) Bpf, 1q : B Ñ pA is symmetric strong monoidal.

(4) f� : pB Ñ pA is symmetric strong monoidal.

Proof. (1) ùñ (2): The assumptions imply that the left Kan extension exists. Thestatement now follows from Lemma 2.6.

(2) ùñ (3): By Proposition 2.8, yA is strong monoidal. Since χf exhibits Bpf, 1qas a pointwise left Kan extension of yA along f , we conclude by the assumption (2) thatBpf, 1q is strong monoidal.


(3) ùñ (4): By Corollary 2.9, yB is exact, and Bpf, 1q is strong monoidal byassumption. But ιf exhibits f� as the pointwise left Kan extension of Bpf, 1q along yB,so again by Lemma 2.6 we conclude that f� is strong monoidal.

(4) ùñ (1): From [34, 35] the unit u of the adjunction f! % f� is the unique 2-cellsatisfying the equation on the left

A pA

pBpA

yA //

f!��

f�oo

��

yA yBf

##χyBf+3

lf +3A pA

pBpA

yA //

f!��

f�oo

��

yA 1pA

{{

id +3u +3�

S pA S pB

pB

pA

pA

Sf! //

�

��

f�

��--1

pA

��

�

f! //

f ! +3

u +3

S pA

S pB

pBpA

S pA

Sf!

��

�

��

f�oo

��

�

��

1S pA

Sf�oo

Su +3

f� +3

�

and f ! and f� determine each other uniquely by the equation on the right. Being theunit of an adjunction, Su is an absolute pointwise left Kan extension, and since f� isassumed to be invertible (4), the common composite of the equation on the right in theprevious display exhibits f� � �B as the pointwise left Kan extension of �A along Sf!.Now paste on the left with yA which is a pointwise left Kan extension by Proposition 2.8.The resulting pointwise left Kan extension can be rewritten as follows:

SA

S pA

S pB

pB

pA

A

pA

SyA

??

Sf!

��

�

��

f��

yA

��

Â��

�

f!

##

1pA

��

;;yA

yA

CK f !CK

id ;C u +3

SA

S pA

S pB

pB

pA

A

pA

B

SyA

??

Sf!

��

�

��

f��

yA

��

Â

��

�

f!

��

??

yA

yB ////f

yA

CKf !CK

lfCK

χyBf+3

SA

S pA

S pB

pB

pA

A

SB

B

SyA

??

Sf!

��

�

��

f��

yA

��

Â

SyB //

Â

��

//Sf

yB //

Bpf,1q

��

//f

SlfCK

f +3 yB +3

χf +3 ιf +3

� �

and since Slf is invertible, we conclude that already

SA S pB

pB

pA

A

SB

B

�

��

f�||""

yA

��

Â

SyB //

Â

��

//Sf

yB //

Bpf,1q

��

//f

f +3 yB +3

χf +3 ιf +3

exhibits f��B as a pointwise left Kan extension of yA�Â

A along SyB�Sf . But already ιf

is a pointwise left Kan extension, and yB is an exact square, so the whole right-hand part


of the diagram is a pointwise left Kan extension. Since furthermore SyB is fully faithful,we can cancel that right-hand part away (e.g. by [27], Theorem 4.47), so in conclusionalso

SA SB

B

pA

A

��

yA

��

Â Â

��

//Sf

Bpf,1q��

//f

f +3

χf +3

,

is a pointwise left Kan extension. It now follows from Lemma 2.2 that f is exact.Note that the implication (4) ùñ (2) was established already by Getzler [19], and

in fact can be extracted from Bunge–Funk [9], Proposition 1.5, as pointed out by theanonymous referee.

2.12. Corollary. For a symmetric monoidal functor τ : SC Ñ M , axiom (2) of beinga regular pattern is equivalent to being exact.

2.13. Morphisms of regular patterns. Recall (from 1.5) that a morphism of regularpatterns is a diagram of symmetric strong monoidal functors

SC1τ1 //

Sf

��

M1

g

��

ω�

SC2 τ2//M2,

where ω is an invertible monoidal natural transformation.

2.14. Proposition. Every such g : M1 ÑM2 is exact.

Proof. The free functor SC1 Ñ SC2 is exact by Corollary 4.6.6 of [38]. The two functorsτ1 and τ2 are exact by assumption, and τ1 is furthermore bijective on objects. It nowfollows from Lemma 2.15 that g is exact.

2.15. Lemma. Given a commutative triangle of symmetric strong monoidal functors

Sf

//

u��

T,

S 1g

??

if f is exact, and u is exact and bijective on objects, then g is exact.


Proof. By Theorem 2.11 it is enough to check that g� is strong monoidal. Consider thecorresponding triangle of pullback functors:

pS pTf�oo

g��pS 1u�

^^

All three functors are lax monoidal; f� and u� are strong monoidal because of exactness.Furthermore u� is monadic since u is bijective on objects, and so u� is conservative. Themonoidal coherences of f� are invertible; but these are obtained by applying u� to thelax coherence of g�. Since u� is conservative we can therefore conclude that already thecoherences for g� must be invertible.

3. The hereditary condition and exactness

In this section we analyse the hereditary condition of Kaufmann and Ward [25] and relateit to Guitart exactness in Proposition 3.3. In Section 5 we shall see that the hereditarycondition is one of two conditions characterising substitudes among pinned monoidalcategories (Proposition 5.12).

3.1. Permutation-monotone factorisation. As in Section 1 for n P N, we denoteby n the linearly-ordered set t1, ..., nu. Any function α : mÑ n factors uniquely as

α � λα � σα,

where σα : m �Ñ m is a permutation that is monotone on the fibre α�1pjq for each j P n,and λα : m Ñ n is monotone1. With reference to α : m Ñ n, if px1, . . . , xmq � pxiqiPm isa sequence of objects, we denote by pxiqαi�j the subsequence consisting of those entrieswhose index maps to j. The order is the induced order on the subset α�1pjq � m.

3.2. The hereditary condition. A symmetric colax monoidal functor τ : SC Ñ Msatisfies the hereditary condition when for all pairs of sequences pxiqiPm and pyjqjPn ofobjects of C, the function

hτ,x,y :°

α:mÑn

±jPn

Mpτpxiqαi�j, τyjq ÝÑMpτpxiqiPm,Â

jPn τyjq

which sends pα, pgjqjPnq to the composite

τpxiqiPm τpxσ�1α iqiPm

ÂjPn τpxiqαi�j

ÂjPn τyj

τσα // τ //

Âj gj//

1This factorisation is not part of a factorisation system, but it is nevertheless very useful.


in M , is a bijection. (Note thatÂ

jPn τpxσ�1α iqλαi�j �

ÂjPn τpxiqαi�j.) Note that the

summation is taken over arbitrary functions α : mÑ n, not just monotone ones.In the case where τ is strict (i.e. when τ is the identity) one has τpxiqiPm �

ÂiPm τxi,

and it may be more convenient to write hτ,x,y as the function°

α:mÑn

±jPn

MpÂ

αi�j τxi, τyjq ÝÑMpÂ

iPm τxi,Â

jPn τyjq

which sends pα, pgjqjPnq to the composite

ÂiPm

τxiσÝÑ

ÂiPm

τxσ�1i �ÂjPn

Âαi�j

τxibjgjÝÑ

ÂjPn

τyj.

In less formal terms, the hereditary condition says that every morphism f of M as onthe right

gj :Â

αi�j τxi ÝÑ τyj f :Â

iPm τxi ÝÑÂ

jPn τyj

can be uniquely decomposed as a tensor product of morphisms gj as on the left, modulosome symmetry coherence isomorphisms in M . A useful slogan for this is: ‘many-to-manymaps decompose uniquely as a tensor product of many-to-one maps’; which expressesthe operadic nature of this condition. The hereditary condition has been discoveredindependently by various people in different guises. While Kaufmann and Ward got itfrom Markl [31] via Borisov and Manin [8], it is also equivalent to the (operad case of the)‘operadicity’ condition of Mellies and Tabareau [32, §3.2].

The main result of this section is

3.3. Proposition. Let τ : SC ÑM be a symmetric colax monoidal functor.

(1) If τ is exact then it satisfies the hereditary condition.(2) If τ is essentially surjective, strong monoidal and satisfies the hereditary condition,

then τ is exact.

and its proof occupies the rest of this section. For (1), we shall first show that thesum-over-functions formula arises from the Day convolution product, and second that thehereditary maps are special cases of the components of a canonical 2-cell associated toτ . For (2), we shall invoke a classical criterion for exactness in terms of a category offactorisations, going back to Guitart himself [21] in some form, and analysed in moredetail in [38].

3.4. Sums over functions from convolution for free symmetric monoidalcategories. Our discussion begins by identifying how sums over functions, as in thedomains of the hereditary condition maps hτ,x,y, arise categorically. For a small categoryC we define the functor

� : SpxSCq ÝÑ xSCto be given on objects as

p�jPn

FjqpxiqiPm �°

α:mÑn

±jPn

Fjpxiqαi�j (4)


where the sum is taken over all functions m Ñ n. We shall now see, as the notationchosen indicates, that � is the Day convolution tensor product.

Let us first exhibit the functoriality of �j Fj in pxiqiPm (that is, verify that the as-

signment in (4) really defines a presheaf on SC). Given pFjqjPn in SpxSCq and a mor-phism pσ, pfiqiq : pxiqiPm Ñ px1iqiPm in SC, note that the permutation σ P Σm restricts toσj : α�1pjq Ñ pασ�1q�1pjq for j P n, and so we get pσj, pfiqαi�jq : pxiqαi�j Ñ px1iqαi�j inSC for each j P n. Thus we define �

jPnFjpσ, pfiqiq as the unique function such that the

square ±jPn

Fjpxiqαi�j p�jPnFjqpxiqiPm

p�jPnFjqpx

1iqiPm

±jPn

Fjpx1iqασ�1i�j

kα //

�jFjpσ,pfiqiq

��

//

kασ�1

��

±jpσj ,pfiqαi�jq

commutes, where kα and kαρ�1 are the sum inclusions. With the functoriality of this

assignment clear by definition, we have thus defined the object map of � : SpxSCq Ñ xSC.We proceed to check that � is functorial. Let pρ, pujqjq : pFjqjPn Ñ pGjqjPn be a morphism

in SpxSCq. For any function α : m Ñ n, pxiqiPm in SC, and j P n, one has the functionpujqpxiqαi�j : Fjpxiqαi�j Ñ Gρjpxiqαi�j. Thus the components of �pρ, pujqjq are defined bythe commutativity of the squares

±jPn

Fjpxiqαi�j p�jPnFjqpxiqiPm

p�jPnGjqpxiqiPm

±jPn

Gρjpxiqαi�j

kα //

�pρ,pujqjqpxiqi

��

//

kρα

��

±jpujqpxiqi

for all α and pxiqi. With the functoriality of this assignment also clear by definition, we

have thus defined the functor � : SpxSCq Ñ xSC.

3.5. Lemma. For any small category C, the functor � : SpxSCq Ñ xSC just defined de-scribes the tensor product for Day convolution on SC.

Proof. The formula (4) is clearly colimit preserving in each Fj, and so it suffices toexhibit an isomorphism

S2C SpxSCq

xSCSC

SySC //

��

//ySC

��

µC �

because then, this isomorphism will exhibit � as a pointwise left Kan extension of ySCµCalong SySC , giving the result by Proposition 2.8. In terms of the notation of Section 2,this isomorphism will then be the natural isomorphism ySC corresponding to our formula(4) for �.


An object of S2C is a sequence of sequences from C, which for convenience we identifyas a pair pψ, pciqiPmq, where ψ : m Ñ n is monotone, and pciqi P SC. Applying ySCµC tothis gives the representable SCp�, pciqiPmq. On the other hand, for pxiqiPl P SC, the set

p� � SpySCqqpψ, pciqiPmqpxiqiPl (5)

is, as with SCppxiqiPl, pciqiPmq, the empty set when l � m. However when l � m, the set(5) is the sum °

α:mÑn

±jPn

SCppxiqαi�j, pciqψi�jq. (6)

To give an element of (6) is to give a function α : m Ñ n, a permutation σ P Σm

such that ψσ � α (which just says that σ restricts to bijections σj : α�1pjq Ñ ψ�1pjq),and for i P m an arrow fi : xi Ñ cσi of C. This is the same as to give a morphismpσ, pfiqiq : pxiqiPm Ñ pciqiPm of SC such that ψσ � α, and this last equation shows thatα is redundant. We thus have our desired isomorphism, whose naturality is very easy tocheck.

3.6. Hereditary condition maps as the components of a natural transfor-mation. We now return to the situation of a general symmetric colax monoidal functorτ : SC ÑM . Denote by θτ the coherence 2-cell datum

SM M

xSCSpxSCq

Â//

Mpτ,1q��

//�

��SpMpτ,1qq θτ +3

for the symmetric lax monoidal functor Mpτ, 1q : M Ñ xSC. By Theorem 2.11, exactnessof τ is equivalent to the invertibility of θτ .

Using the explicit description of the Day convolution tensor product in SpxSCq, justestablished in Lemma 3.5, we see that the components of θτ at pwjqjPn in SM amount tomaps of sets

pθτpwjqjqpxiqi :°

α:mÑn

±jPn

Mpτpxiqαi�j, wjq ÝÑMpτpxiqi,Âj

wjq

for pxiqiPm in SC, which we proceed to describe.

3.7. Lemma. The component pθτpwjqjqpxiqi is the function which sends pα, pgjqjPnq to thecomposite

τpxiqiPm τpaσ�1α iqiPm

ÂjPn τpxiqαi�j

ÂjPn τpwjq

τσα // τ //

Âj gj

// (7)

(Note that in the special case where wj � τyj (that is, the components of θτ in the imageof τ), we recover precisely the hereditary maps hτ,x,y.)


Proof. We saw in Section 2 that Mpτ, 1q is the pointwise left Kan extension of ySC alongτ . It then follows from Theorem 2.4.4(1) of [38] that θτ can be characterised as the uniquenatural transformation satisfying the equation

S2C SM

M

xSC

SC SpxSCq

Sτ //

Â

��

Mpτ,1qyy%%

ySC

��

µC

��SySC ��

SMpτ,1q

�

��

ySC+3 θτ +3

Sχτ+3

�

S2C SM

M

xSC

SC

Sτ //

Â

��

Mpτ,1q��

ySC

��

µC

τ //

τ +3

χτ +3

(8)

To prove the lemma it is therefore enough to check that the stated formula for θτ satisfiesEquation (8). As in the proof of Lemma 3.5 we denote an object of S2C as a pairpψ, pciqiq, where ψ : m Ñ n is monotone and pciqiPm is an object of SC. We check thatthe pψ, pciqiq-components of either side of (8) agree. Note that for pxiqiPl in SC, theppxiqiPl, pψ, pciqiqq-components of both sides of (8) give functions

SCppxiqiPl, pciqiPmq ÝÑMpτpxiqiPl,Â

j τpciqψi�jq.

The domain of these functions is empty when l � m, and so it suffices to considerthe case when l � m. Let pρ, pfiqiPmq be in SCppxiqiPm, pciqiPmq. Note that for j P n,the permutation ρ restricts to bijections ρj : pψρq�1pjq Ñ pψq�1pjq, and so one haspρj, pfiqψi�jq in SCppxiqψρi�j, pciqψi�jq. Applying the ppxiqi, pψ, pciqiqq-component of theleft hand side of (8) to pρ, pfiqiPmq gives the composite (7) with gj � τpρj, pfiqψi�jq.On the other hand, applying the ppxiqi, pψ, pciqiqq-component of the right hand side topρ, pfiqiPmq gives the composite τ pψ,pciqiq � τpρ, pfiqiq. These coincide by the naturality ofτ .

Proof of Proposition 3.3 (1). In view of Lemma 3.7, the hereditary condition for τsays that the natural transformation θτSpτηCq is invertible. The result follows since byTheorem 2.11, exactness is equivalent to the invertibility of θτ .

The argument just given is not immediately adaptable to prove the converse. Even if τis essentially surjective, the functor SpτηCq : SC Ñ SM is not, and so even for essentiallysurjective τ , it is not apparent that the invertibility of θτSpτηCq implies the invertibilityof θτ . Hence the need for the further assumption on τ of strong monoidality among thehypotheses of Proposition 3.3(2). Moreover, it seems to us that the clearest proof of thisresult follows from the consideration of factorisation categories, to which we now turn.

3.8. Factorisation category of a morphism f : Fv Ñ bkwk. Consider a symmet-ric colax monoidal functor F : V Ñ W (with coherences F : pbiviq Ñ biFvi). For eachmorphism in W of the form f : Fv Ñ bkwk, we define the category of factorisations


Factpfq as having objects diagrams of the form

Fvf

//

Fh $$

bkwk.

F pbkukq // bkF pukq

bkgk

99

Morphisms in Factpfq are connecting maps (in V ) between the uk and the u1k making aleft-hand triangle commute already in V , and making the right hand triangle commutein W . (The middle square commutes by naturality of the coherence data.)

By Proposition 4.5.5(1) of [38], F is exact as a symmetric colax monoidal functor(� colax morphism of S-algebras) if and only if it is exact as a colax monoidal functor(� colax morphism of M-algebras). Applying Lemma 2.4 to the lax square containing F ’scolax M-morphism datum, gives the following explicit characterisation of exact symmetriccolax monoidal functors.

3.9. Lemma. A symmetric colax monoidal functor F : V Ñ W as above is exact if andonly if for every f : v Ñ bkwk the category Factpfq is connected.

When V is SC, and F is strict monoidal τ : SC Ñ M , the description of Factpfqsimplifies considerably. In this case v � pxiqiPm, and each uk is a sequence pziqiPmk .Furthermore, since h is a map in SC it is a labelled permutation, so in particular theconcatenated sequence ppziqiPmkqkPr is of the same length as m, so altogether we can writea factorisation of f : τpxiqiPm Ñ bkPrwk as

τpxiqiPmτphq

// τppziqβi�kqkPr � bkPr bβi�k τzibkPrgk

// bkPrwk,

where β : m Ñ r is monotone. Since h is a labelled permutation, we can keep the purepermutation part as a first factor, and then absorb the second factor (the ‘labels’) intothe gk on the right. Renaming those gk accordingly, we arrive at a factorisation

τpxiqiPm� // τppxiqβi�kqkPr � bkPr bβi�k τxi

bkPrgk// bkPrwk.

Factorisations of this form we call normalised. Clearly every factorisation receives amorphism from its normalisation.

Proof of Proposition 3.3 (2). By strictification, we can assume that τ is strictmonoidal and identity on objects. We fix f : τpxiqiPm � biPmτxi Ñ bkPrwk and aim toshow that Factpfq is connected. The identity-on-object condition means that each wkis a tensor product of certain τyj, say wk � bγj�kτyj, where γ : n Ñ r is monotone.Altogether,

bkPrwk � bkPr bγj�k τyj � bjPnτyj.

The map f : biPmτxi Ñ bkPrwk is now of the form

f : biPmτxi Ñ bjPnτyj,


and by the hereditary condition we get a factorisation as

biPmτxi� // bjPn bαi�j τxi

bjPnsj// bjPnτyj,

which can also be written as

biPmτxi� // bkPr bγj�k bαi�jτxi

bkPrpbγj�ksjq// bkPrpbγj�kτyjq. (9)

This we refer to as the standard factorisation of f . It is seen to be an object in Factpfqby putting gk � bγj�ksj. In particular we have now shown that Factpfq is not empty.

Given now any other normalised object in Factpfq

biPmτxi� // bkPr bβi�k τxi

bkPrg1k// bkPrwk, (10)

where β : mÑ r monotone, we would like to connect it to the standard factorisation justconstructed. To this end we apply the hereditary condition to each of the maps

g1k : bβi�kτxi Ñ wk � bγj�kτyj.

This gives us a function αk : β�1pkq Ñ γ�1pkq, and all these functions assemble into afunction α : m Ñ n such that γ � α � β. With reference to these αk, the hereditarycondition gives us a normalised factorisation of gk as

bβi�kτxi� // bγj�k bαi�j τxi

bγj�ks1j// bγj�kτyj

so that altogether f factors as

biPmτxi� // bkPr bβi�k τxi

� // bkPr bγj�k bαi�jτxibkPrbγj�ks

1k // bkPr bγj�k τyj.

If we take the middle permutation to belong to the left-hand factor, then we obtain afactorisation of the standard shape (9), and by the uniqueness property in the hereditarycondition, this must actually be equal to the standard factorisation (9), that is s1j � sjfor all j P n. On the other hand, if we let the middle permutation belong to the right-hand factor, we get precisely the given normalised factorisation (10). Hence the givennormalised factorisation is connected to the standard factorisation. Since we alreadyremarked that any factorisation is connected to its normalisation, we have altogethershown that Factpfq is connected.

An alternative proof can be derived from the results in Section 5: by Proposition 5.12,we may assume that τ is of the form Fpφq, where φ : C Ñ P is a substitude. It can bechecked directly that for any morphism of operads φ, the symmetric monoidal functorFpφq is exact. See [38] (Corollary 3.4.1) for a proof.


4. Kaufmann–Ward comma-category condition

4.1. Feynman categories. Kaufmann and Ward [25] define a Feynman category to bea symmetric strong monoidal functor τ : SC ÑM satisfying the three conditions

(1) C is a groupoid(2) τ induces an equivalence of groupoids SC �ÑMiso

(3) τ induces an equivalence of groupoids SpMÓCqiso�Ñ pMÓMqiso.

In this section we show that the comma-category condition (3), can be reformulated interms of the hereditary map

h :� hτ,x,y :°

α:mÑn

±jPn

MpÂ

αi�j τxi, τyjq ÝÑMpÂ

iPm τxi,Â

jPn τyjq

from 3.2. This is more or less implicit in [25]. This section does not seem to generalise tothe enriched setting.

4.2. Proposition. For an essentially surjective symmetric strong monoidal functor τ :SC ÑM , the following are equivalent:

(1) τ : SC ÑM is hereditary and τiso : SCiso ÑMiso is an equivalence of groupoids(2) The natural map τcomma : SpMÓCqiso Ñ pMÓMqiso is an equivalence of groupoids.

In view of Proposition 3.3, this shows:

4.3. Corollary. A Feynman category is precisely a regular pattern τ : SC Ñ M forwhich C is a groupoid and τiso : SCiso ÑMiso is an equivalence of groupoids.

Proof of Proposition 4.2. We immediately reduce to the strict situation, where τ isidentity-on-objects. Lemma 4.4 below says that if just τcomma is fully faithful, then alsoτiso is fully faithful (and therefore actually an isomorphism). So we can separate that outas a global assumption. Now τcomma is full if and only if h is injective by Lemma 4.6, andτcomma is essentially surjective if and only if h is surjective by Lemma 4.5. Finally, it isactually automatic that τcomma is faithful (again by Lemma 4.4).

4.4. Lemma. If τcomma : SpMÓCqiso Ñ pMÓMqiso is full, respectively faithful, then τiso :SCiso Ñ Miso is full, respectively faithful. Conversely, if τiso is faithful then τcomma isfaithful.

Proof. The first statements follow immediately from the commutative diagram

SCisoτiso //

��

Miso

��

SpMÓCqiso τcomma// pMÓMqiso

since the vertical maps are fully faithful.


For the last statement, the isomorphisms in the image of τcomma are of the form

Âj bixi

a //

bjgj

��

Âj bix

1i

bjg1j

��

bjyj b// bjy

1j.

If individually the horizontal isos can arise from SC in at most one way, then takentogether it is even harder to arise from SpMÓCq.

4.5. Lemma. For an identity-on-objects symmetric strict monoidal functor τ : SC ÑM ,such that SCiso �Miso, the following are equivalent:

(1) The map h in the hereditary condition is surjective.(2) The natural map τcomma : SpMÓCqiso Ñ pMÓMqiso is essentially surjective.

Proof. Throughout the proof we suppress τ on objects, since anyway τ is identity-on-objects.

We first prove that if h is surjective, then τcomma is essentially surjective. Givensome object in MÓM , that’s precisely an element in the codomain of h, say f : bixi Ñbjyj. By surjectivity of h, there is an element pα, g1, . . . , gnq in the domain of h, whosetensor product is f . Now just the sequence pg1, . . . , gnq is an object in SpMÓCq and byassumption, their tensor product is isomorphic to f (by the permutation σα obtained fromα).

Conversely, assuming τcomma is essentially surjective, let us prove that h is surjective.Given f : bixi Ñ bjyj, an element in the codomain of h, we need to construct an elementon the left—that’s a tuple pα, g1, . . . , gnq—such that the composite

bixiσα // bj bk xk

bjgj// bjyj

is equal to f . (Here σα is the permutation part of α, obtained by permutation-monotonefactorisation). Since τcomma is essentially surjective, there exists an object pλ1, h11, . . . , h

1nq

in SpMÓCq (here λ1 : mÑ n is monotone, and h1j : bλ1i1�j1x1i1 Ñ y1j1) whose tensor product

is isomorphic to f :

bixi �a //

f

��

bj1 bλ1i1�j1 x1i1

bj1h1j1

��

bjyj�

b// bj1y

1j1 .

Since τiso is full, both a and b come from SC, and in particular can be written as apermutation followed by a tensor product of isos in C. Let ς be the permutation underlyinga and let ρ be the permutation underlying b. Put

α :� ρ�1 � λ1 � ς.


The permutation ρ : n �Ñ n is such that with j1 � ρj we have yj � y1j1 . Conjugating withthis isomorphism we find

bj bλ1i1�ρj x1i1

b //

bjg1j

��

bj1 bλ1i1�j1 x1i1

bjh1j1

��

bjyj b// bj1y

1j1 .

(The permutation ρ : m �Ñ m underlying b permutes the blocks according to the per-mutation b.) Except for some isos in C, the new map g1j is essentially h1ρj. The tensorproduct of the new maps g1j now have underlying indexing map λ :� ρ�1 � λ1 � ρ, which isdifferent from λ1, but is still monotone. On the other hand, the permutation ς : m �Ñ mis such that with i1 � ςi we have xi � x1i1 . Using this, we can rewrite the upper left-handcorner

bj bλi1�ρj x1i1 � bj bλςi�ρj xi � bj bαi�j xi,

but a little care is needed with this substitution, since it may permute stuff inside eachj-factor. However, this permutation can be absorbed into each g1j and now called gj (andthis does not affect λ), giving altogether

bixiσ //

f

��

bj bαi�j xi

bjgj

��

// bj1 bλ1i1�j1 x1i1

bj1h1j1

��

bjyj �// bjyj // bjy

1j1 .

The remaining permutation σ is monotone on λ-fibres by construction, and since α � λ�σ,we see that pα, g1, . . . , gnq is a solution to our problem: by construction, h applied topα, g1, . . . , gnq is the original f . Hence h is surjective.

4.6. Lemma. For an identity-on-objects symmetric strict monoidal functor τ : SC ÑM ,such that SCiso �Miso, the following are equivalent:

(1) The map h in the hereditary condition is injective.(2) The natural map τcomma : SpMÓCqiso Ñ pMÓMqiso is full.

Proof. Throughout the proof we suppress τ on objects, since anyway τ is identity-on-objects.

Let us show that if τcomma is full then h is injective. Suppose we have two elements inthe domain of h both giving f . Say pα, gjq and pα1, g1jq. (In both cases j runs to the samen: that’s part of the data in f). This gives us now a commutative diagram of maps in M :

Âj bixi

bjgj

��

bixiσoo

f

��

σ1 //Â

j bixi

bjg1j

��

bjyj bjyj bjyj


The outer vertical maps, bjgj and bjg1j, are now two objects in MÓM , exhibited iso-

morphic by means of σ1 � σ�1 and the identity at the bottom. Since τcomma is full, thisisomorphism comes from one in SpMÓCq, hence is given by a labelled permutation of n.Since the bottom map is the identity, also the permutation σ1 � σ�1 is the identity on theouter tensor factors, those indexed by j. So σ and σ1 agree on outer factors. But they arealso monotone on fibres, so in fact they must agree completely. It follows that gj � g1j,and hence in particular also that α � α1.

Conversely, assuming that h is injective, let us show that τcomma is full. Given twoobjects in MÓM in the image of SpMÓCq, pictured vertically, and an iso between them(consisting of two isomorphisms, pictured horizontally):

Âj bixi

a //

bjgj

��

Âj bix

1i

bjg1j

��

bjyj b// bjy

1j

A priori, we don’t know that the j run to the same n, but in fact they do, because ofthe existence of b: since τiso is fully faithful, b is in fact the image of an isomorphism inSC, which is to say that it is a labelled permutation. For the same reason, a is a labelledpermutation too. Since the vertical maps are in the image of τcomma, their correspondingα and α1 have trivial permutation part. It follows that the permutation a must actuallybe a refinement of the permutation b. In particular we have necessarily α � α1, which isa monotone map, since it just comes from the tensor product.

It remains to check that a and b together actually form a valid isomorphism in SpMÓCq.We have found that the original square is the tensor product of a sequence of squares ofthe form

biPα�1σ�1jxia //

gj

��

biPα�1pjqx1i

g1j��

yσ�1j b// y1j

It remains to check that each of them commutes. For fixed j P n, the two composites inthis individual square are both elements in the set

Mpâ

iPα�1σ�1pjq

xi, y1jq,

so altogether they form a tuple which is an element in¹jPn

Mpâ

iPα�1σ�1pjq

xi, y1jq.

That’s in the σ � α summand of the domain of h. If one of the squares did not commute,it would thus constitute two distinct elements in this set, with the same image undertensoring (the map h). Since h is injective, we conclude that in fact all those squares docommute, and hence form a valid isomorphism in SpMÓCq, as required.


5. Hermida-type adjunctions and the main theorem

5.1. Operads. By operad we always mean symmetric coloured operad (in Set), alsoknown as symmetric multicategory. Hence, an operad P consists of a set I of objects (alsocalled colours), for each pair px1, ..., xn; yq consisting of a finite sequence of input objectspx1, ..., xnq and one output object y, a set P px1, ..., xn; yq of operations from px1, . . . , xnqto y. Moreover there is an identity operation 1x : pxq Ñ x in P px;xq, and a substitutionlaw satisfying the usual axioms.

We shall use various shorthand notation for sequences px1, . . . , xnq, such as pxiqiPn, orjust pxiqi, or even x, when practical.

Operads form a 2-category Opd, the morphisms being morphisms of operads in theusual sense. A 2-cell

P Q

f

''

g

77ω��

consists of components ωx : pfxq Ñ gx, required to be natural with respect to all opera-tions of P , that is, given an operation b : pxiqi Ñ y, one has ωyfpbq � gpbqpωxiqi.

A small category can be regarded as an operad with only unary operations, and inthis way Cat becomes a coreflective sub-2-category of Opd (the coreflector picks outthe unary part of an operad.) Various notions from category theory makes sense alsofor operads: in particular, a morphism of operads f : P Ñ Q is called fully faithful if itinduces bijections on multihomsets P px; yq �Ñ Qpfx; fpyqq. Gabriel factorisation worksthe same for operads as for categories, as exploited also in [18]: every operad morphismfactors as bijective-on-objects followed by fully faithful. Just as in the category case, thisis an enhanced factorisation system. And just as in the category case, one can alwayschoose the bijective-on-objects part to be actually identity-on-objects (with which choicethe factorisation is unique).

5.2. Hermida adjunctions. For any symmetric monoidal category M , its endomor-phism operad EndpMq has objects those of M , and sets of operations given by

EndpMqppx1, ..., xnq; yq � MpÂn

i�1 xi, yq.

The category P -AlgpMq, of P -algebras in M , is defined as

P -AlgpMq :� OpdpP,EndpMqq

which at the level of objects, says that a P -algebra in M is a morphism of operadsP Ñ EndpMq. The assignment M ÞÑ EndpMq is the effect on objects of a 2-functorEnd : SMC Ñ Opd, where SMC denotes the 2-category of symmetric monoidal cate-gories, symmetric strong monoidal functors, and monoidal natural transformations. Be-low, we will also consider the restriction of this 2-functor to a 2-functor SMCs Ñ Opd,where SMCs is the locally full sub-2-category of SMC consisting of the symmetric strictmonoidal categories and symmetric strict monoidal functors.


In the other direction, one can associate to any operad P , a symmetric strict monoidalcategory FP , which has the following explicit description. An object of FP is a finitesequence of objects of P . A morphism px1, ..., xmq Ñ py1, ..., ynq in FP , consists of anindexing function α : m Ñ n, together with for each j P n, an operation bj : pxiqαi�j Ñyj. The category structure of FP comes from substitution in P and the composition of(indexing) functions. Thus the homs of FP are given by

FP�pxiqiPm, pyjqjPn

��¸

α:mÑn

¹jPn

P ppxiqαi�j; yjq.

Note that for a category C regarded as an operad, we have a canonical identificationFC � SC.

Now, FP enjoys a strict universal property amongst all symmetric strict monoidalcategories, expressed by the isomorphisms of categories on the left

SMCspFP,Mq � OpdpP,EndpMqq SMCpFP,Mq � OpdpP,EndpMqq

2-naturally in M ; and also a bicategorical universal property amongst all symmetricmonoidal categories expressed by the equivalences of categories on the right, which arepseudo-natural in M . Taken together one thus has a 2-adjunction as indicated on the left

SMCs OpdEnd

//

FooK SMC Opd

End//

FooKb

and a biadjunction as indicated on the right. We shall call these the Hermida adjunctionsin honour of Claudio Hermida, who studied the strict version in the non-symmetric case[22] (see also [16], Theorem 4.2). In [37] Corollary 6.4.7, they were obtained formally fromthe 2-monad S.

Recall that to give a biadjunction with left adjoint F and right adjoint End is to givepseudo-natural transformations ηH : 1Opd Ñ EndF and εH : FEnd Ñ 1SMC togetherwith invertible modifications

F FEndF

F

FηH//

εHFzz$$1F

�End EndFEnd

End

ηH End//

End εHzz$$1End

�

sometimes called the left and right triangulators for the biadjunction. In our case the2-adjunction can be recovered from the biadjunction by restricting from SMC to SMCs.In terms of the unit and counit, this says the following: (1) the unit of the Hermidabiadjunction is the same as that of the 2-adjunction and so is strictly 2-natural; (2) theleft triangulator is an identity; (3) for morphisms of SMCs the associated εH pseudo-naturality datum is an identity; and (4) for objects of SMCs the associated componentof the right triangulator is an identity. We turn now to an explicit description of thecomponents of ηH and εH .


The counit component at M P SMC

εHM : FpEndpMqq ÝÑM pxiqi ÞÑÂ

i xi

has object map as indicated, and hom functions

FpEndpMqq�pxiqiPm, pyjqjPn

�ÝÑM

�ÂiPm xi,

ÂjPn yj

�

which sendpα : mÑ n, pgj :

Âαi�j xi Ñ yjqjq

to the composite ÂiPm

xiσÝÑ

ÂiPm

xσ�1i �ÂjPn

Âαi�j

xibjgjÝÑ

ÂjPn

yj

where σ � σα is the permutation part of α (given by the permutation/monotone fac-torisation), and the unnamed isomorphism is obtained from the coherences for M , thesebeing identities when M is strict.

The component of the unit at P P Opd is given on objects as

ηHP : P ÝÑ EndpFP q

x ÞÝÑ pxq.

More interesting is to see what ηH does on sets of operations: it is a map

P px1, . . . , xm; yq Ñ EndpFP q�px1q, . . . , pxmq; pyq

�

but the last set we can unravel as

� FP�px1, . . . , xmq, pyq

�

since the tensor product in FP is just concatenation of sequences;

�¸

α:mÑ1

¹jP1

P ppxiqαi�j; yjq

by definition of hom sets in FP ;

� P px1, . . . , xm; yq

since there exists only one indexing map m Ñ 1. In the end, ηHP is the identity map onoperations. In conclusion:

5.3. Lemma. The components of ηH , the unit for the Hermida adjunction, are fully faith-ful operad maps.


5.4. Pinned operads. We use the term pinned for an object (e.g. a symmetric monoidalcategory, an operad, or a substitude) equipped with a map singling out some objects (thepins).

A pinned operad is a triple pC, φ, P q, in which C is a category regarded as an operadwith only unary operations, P is an operad, and φ : C Ñ P is a morphism of operads.Pinned operads assemble into a 2-category pOpd. A morphism pC1, φ1, P1q Ñ pC2, φ2, P2qis a triple pf, g, ωq consisting of a functor f : C1 Ñ C2, an operad morphism g : P1 Ñ P2,and an invertible 2-cell φ2f Ñ gφ1 in Opd. A 2-cell pf, g, ωq Ñ pf 1, g1, ω1q is a pair pα, βq,where α : f Ñ f 1 is a natural transformation, and β : g Ñ g1 is a 2-cell of Opd, such that

C1 P1

P2C2

φ1//

g1

��//

φ2

��f f 1

��

ω1 +3α +3 �

C1 P1

P2C2

φ1//

g1

��//

φ2

��f g

��

ω +3 β +3

in Opd. Compositions for pOpd are inherited from Opd in the obvious way. A mor-phism pf, g, ωq of pOpd is said to be strict when ω is an identity 2-cell, and the wideand locally full sub-2-category of pOpd consisting of the strict morphisms is denotedpOpds. The 2-categories pSMC and pSMCs of pinned symmetric monoidal categories,and pinned symmetric strict monoidal categories, were described above in 1.4.

5.5. Pinned Hermida adjunctions. The Hermida adjunctions have pinned analogues

pSMCs pOpdsEndp

//

Fpoo

K pSMC pOpdEndp

//

Fpoo

Kb

which we now describe. The object maps of Endp and Fp are

SC Mτ // � // C EndpSCq

ηHC // EndpMqEndpτq

// C Pφ// � // SC FP

Fpφq//

respectively, and since End and F are 2-functors and ηH is 2-natural, these definitionsextend in the obvious way to arrows and 2-cells.

The component of the unit ηp of Fp %b Endp at pC, φ, P q is p1C , ηHP , idq, as depicted

in the diagram on the left

C P

EndpFP qEndpSCqC

φ//

ηHP��

//

EndpFpφqq//

ηHC

SC FpEndpSCqq FpEndpMqq

MSC

FpηHC q//FpEndpτqq

//

εHM��//

τ

εHSCxx�εHτ

which commutes by the naturality of ηH . So the components of ηp all live in pOpds. Thecomponent of the counit εp of Fp %b Endp at pC, τ,Mq is p1C , ε

HM , ε

Hτ Fpη

HC qq, as depicted


on the right in the previous display. Note that when pC, τ,Mq is strict, the pseudo-naturality datum εHτ is an identity, and then this component of εp lives in pOpds. Thepseudo-naturality data for the unit and counit, and the left and right triangulators forFp %b Endp are inherited from F %b End. The manner in which the biadjunction restrictsto a 2-adjunction is clearly inherited also.

5.6. Substitudes [14]. The notion of substitude was introduced by Day and Street [14]as a general setting for substitution. It is a common generalisation of operad and (sym-metric) monoidal category. See the appendix of [5] for a concise account of the basictheory. A substitude is like an operad, but allowing for a category of objects instead ofjust a set of objects. The data is

– a category C– a functor P : pSCqop � C Ñ Set– composition and unit laws, subject to non-surprising axioms.Substitudes can be described also as monads in the bicategory of generalised species [17].

For the present purposes, the most convenient is to package the definition into the follow-ing (cf. [15, 6.3] for the equivalence between the two formulations of the definition): Asubstitude is a pinned operad φ : C Ñ P in which φ is the identity on objects. We denoteby Subst the full sub-2-category of pOpd consisting of the substitudes, and denote byJ : Subst Ñ pOpd the inclusion.

5.7. Substitude coreflection. Since the bijective-on-objects morphisms form the leftclass of an enhanced factorisation system (Gabriel factorisation), by general principles,the inclusion functor J : Subst Ñ pOpd has a right adjoint, denoted p�q1, forming a2-adjunction

pOpd Subst.p�q1

//

JooK

Explicitly, given a pinned operad pC, φ, P q, factoring φ as identity-on-objects then fullyfaithful as on the left in

C P

P 1

φ//::

εφ$$φ1

C1 P 11 P1

P2P 12C2

φ11 //εφ1 //

g��//

εφ2//

φ12

��f f 1

�� ω1

one obtains the substitude pC, φ1, P 1q together with the pC, φ, P q-component of the counitof J % p�q1, which we denote as εφ. Given a morphism pf, g, ωq : pC1, φ1, P1q Ñ pC2, φ2, P2qof pOpd, one induces f 1 and ω1 uniquely so that the composite on the right in the previousdisplay is ω, using the enhancedness of Opd’s Gabriel factorisation. Using the fact thatthis Gabriel factorisation is also Cat-enriched, one can easily exhibit p�q1’s 2-cell mappingexplicitly.


It is suggestive to write the operad part of φ1 : C Ñ P 1 as P |C: it is the operadbase-changed to its pins.

Cφ

//

φ1 !!

P

P |C

==

The operad P |C has the same objects as C by construction, and operations

P |Cpx1, . . . , xm; yq � P pφx1, . . . , φxm;φyq.

5.8. The substitude biadjunction. Now compose the pinned Hermida biadjunction5.5 with the coreflection of substitudes into pinned operads 5.7:

pSMC pOpd SubstFp

//

Endpoo

Kb

p�q1//

JooK

The composed biadjunction

pSMC SubstEndiop

//

Fiopoo

Kb (11)

goes like this: the left adjoint takes a substitude C Ñ P to the pinned symmetric monoidalcategory SC Ñ FP , and the right adjoint takes a pinned symmetric monoidal categorySC ÑM to C Ñ EndpMq|C.

Unlike the Hermida biadjunction and the pinned version, this one has invertible unit:

5.9. Proposition. The unit for the substitude biadjunction Fiop % Endiop is invertible.

Proof. Let φ : C Ñ P be a substitude. In the diagram

Cφ

//

φ,,

P

ηHP

��

P

��

CηHC //

io ++

EndpSCqEndpFpφqq

// EndpFpP qq

EndpFP q|Cff

55

the back rectangle is the unit ηpφ. The unit we are after, ηiop

φ , is obtained by Gabrielfactorising its horizontal arrows: this induces the dotted arrow by functoriality of thefactorisation, and the left-hand square is now ηiop

φ . It is clearly invertible if and only ifthe dotted arrow is invertible. But the dotted arrow is invertible because it compares twoGabriel factorisations of EndpFpφqq � ηHC : on one hand φ followed by ηHP (the latter beingfully faithful by Lemma 5.3) and on the other hand the factorisation at the bottom of thediagram.


5.10. Counit of the substitude biadjunction. Thanks to Proposition 5.9 the 2-category Subst is biequivalent to the full sub-2-category of pSMC , spanned by thosepinned symmetric monoidal categories pC, τ,Mq for which the counit εiop

τ is an equiv-alence. In the remainder of this section, we determine for which pC, τ,Mq this is thecase. Fundamental to this characterisation is the hereditary condition 3.2. We begin thediscussion by characterising equivalences in pSMC .

5.11. Lemma. A morphism pf, g, ωq : pC1, τ1,M1q Ñ pC2, τ2,M2q of pSMC is an equiv-alence if and only if the functors f and g are equivalences of categories.

Proof. It suffices to show that the forgetful 2-functor

pSMC ÝÑ Cat �Cat pC, τ,Mq ÞÑ pC,Mq

reflects equivalences. So we suppose that pf, g, ωq : pC1, τ1,M1q Ñ pC2, τ2,M2q is a mor-phism of pSMC such that f and g are equivalences of categories. Choose adjoint pseudoinverses f 1 and g1 of f and g respectively, and then note that g1 admits a unique symmetricstrong monoidal structure making the adjoint equivalence it participates in in Cat, intoone in SMC . Thus we have adjoint equivalences pSf, Sf 1q and pg, g1q in SMC , and soone can take the mate ω1 : τ1Spf

1q � g1τ2 of ω. This makes pf 1, g1, ω1q into a morphismof pSMC . Since ω and ω1 are mates via the adjoint equivalences pSf, Sf 1q and pg, g1q,these assemble to an adjoint equivalence in pSMC , exhibiting pf 1, g1, ω1q as an adjointpseudo-inverse of pf, g, ωq.

We now unpack the counit component εiopτ for pC, τ,Mq in pSMC . First we describe

the substitude Endioppτq. It is obtained by Gabriel factorising the pinned operad Endppτq,which is the top composite here:

CηHC //

Endioppτq $$

EndpSCqEndpτq

// EndpMq

EndpMq|C

εff

88

Note that the operad EndpMq|C has objects those of C, and operation sets

EndpMq|C�x1, . . . , xm; y

�� EndpMq

�τx1, . . . , τxm; τy

��M

�τx1 b � � � b τxm, τy

�.

Now apply Fiop, rendered as the top square in the next diagram; the diagram as a wholeis the counit εiop

τ , the key part being of course the right-hand vertical composite:

SC FpEndpMq|Cq

FEndpMq

MSC

SC FEndpSCq

FEndioppτq//

Fpεq

��

εHM��//

τ

FηHSC //FEndpτq

//

εHSCyy

�εHτ

(12)


5.12. Proposition. For a pinned symmetric monoidal category τ : SC ÑM , the counitcomponent εiop

τ is an equivalence if and only if τ is essentially surjective and satisfies thehereditary condition 3.2.

Proof. By Lemma 5.11, εiopτ is an equivalence in pSMC if and only if εHMFpεq is an

equivalence of categories. Since FEndioppτq is an identity on objects, Diagram (12) showsthat εHMFpεq is essentially surjective if and only if τ is. So we have reduced to the casewhere τ is essentially surjective, which is the content of the next lemma.

5.13. Lemma. An essentially surjective pinned symmetric monoidal category τ : SC ÑM satisfies the hereditary condition if and only if εHMFpεq is fully faithful.

Proof. Applying Power coherence, we can reduce to the case where τ is an identity-on-objects symmetric strict monoidal functor. Let us now unpack the effect on morphisms ofεHMFpεq in this case. The objects of the symmetric monoidal category FpEndpMq|Cq aresequences of objects in C, and εHMFpεq sends px1, . . . , xmq to τx1 b � � � b τxm. The homsets of FpEndpMq|Cq are

FpEndpMq|Cq�pxiqiPm, pyjqjPn

��¸

α:mÑn

¹jPn

M� âαi�j

τxi, τyj�,

and the hom mapping of εHMFpεq into Mpbiτxi,bjτyjq, sends pα, g1, . . . , gnq to the com-posite Â

iPm

τxiσÝÑ

ÂiPm

τxσ�1i �ÂjPn

Âαi�j

τxibjgjÝÑ

ÂjPn

τyj.

Thus, the hom functions for εHMFpεq are exactly the functions hτ,x,y whose bijectivity isthe hereditary condition.

Taking Theorem 2.11 and Propositions 3.3 and 5.12 together, we have thus arrived atthe first part of the main theorem:

5.14. Theorem. Fiop : Subst Ñ pSMC induces a biequivalence Subst � RPat.

5.15. Operads. There are three ways of regarding operads as substitudes. For a givenoperad P , the options are: the trivial pinning, where C � objpP q, the discrete category ofobjects; the canonical groupoid pinning, where C � P iso

1 , the groupoid of invertible unaryoperations; and the full pinning, where C � P1, the category of all unary operations.

We are interested at the moment in the canonical groupoid pinning, P iso1 Ñ P . Note

that the functor p�qiso : Cat Ñ Grpd is not a 2-functor. However it does make senseto apply p�qiso to invertible natural transformations, and so p�qiso is Grpd-enriched. If,for any 2-category K, we denote by pKq2-iso its underlying groupoid-enriched category,obtained from K simply by ignoring non-invertible 2-cells, then another way to expressthese considerations is to say that one has a 2-functor p�qiso : pCatq2-iso Ñ Grpd.

Thus, the process of taking the canonical groupoid pinning is the effect on objects ofa 2-functor

G : pOpdq2-iso ÝÑ Subst


which is 2-fully-faithful when the codomain is restricted to pSubstq2-iso. Writing FeynCatfor the full sub-2-category of ppSMCq2-iso consisting of the Feynman categories in thesense of Kaufmann and Ward, the second part of our main theorem is as follows.

5.16. Theorem. Fiop � G restricts to a biequivalence pOpdq2-iso � FeynCat.

Proof. First we check that a pinned symmetric monoidal category in the image is aFeynman category, using 4.3. We already know that it satisfies the hereditary condition,and by construction P iso

1 is a groupoid. It remains to check that

SP iso1 Ñ pFP qiso

is an equivalence. These two categories have the same objects, namely sequences of objectsin P . The arrows in FP from x to y form the hom set

¸α:mÑn

¹jPn

P ppxiqiPα�1pjq; yjq

They are composed as operations in P . There is an obvious forgetful functor to Set, givenby returning the indexing set for the sequence. For an arrow to be invertible, at leastits underlying α must be a bijection, and furthermore it is then clear that the involvedoperations have to be invertible unary operations.

Conversely, if we start with a Feynman category τ : SC Ñ M , the image in Substis the substitude C Ñ EndpMq|C, and since C is a groupoid and SC � Miso, it is clearthat the invertible unary operations of EndpMq|C are precisely those of C, which is thecondition for the substitude to be in the image of G.

5.17. Two other subcategories of RPat equivalent to the category ofoperads. Let us note that there are two other subcategories of RPat which are equivalentto the category of operads, given by the two other natural embeddings of Opd into Subst:one takes an operad P to its discrete pinning objpP q Ñ P—this is a 1-functor only, as itis not defined on 2-cells. The other embedding takes P to its full pinning P1 Ñ P (this isa honest 2-functor).

The first corresponds to the subcategory of discrete substitudes, which in turn corre-sponds to the subcategory of regular patterns SC Ñ M for which C is discrete. Whilethis is obviously a stronger condition than the first Feynman-category condition, that ofbeing a groupoid, on the other hand the second Feynman-category condition, SC �ÑMiso

is not in general satisfied. Imposing this second condition on top of the discreteness con-dition gives the notion of discrete Feynman category, mentioned in [25] to be related tooperads. More precisely this notion corresponds to operads in which there are no unaryoperations other than the identities (the locus of operads for which the groupoid-pinningand discrete-pinning embeddings coincide). (This embedding of the 1-category of operadsinto regular patterns was also observed by Getzler [19].)

The second embedding of operads into substitudes, endowing an operad P with its fullpinning P1 Ñ P , has as essential image that of normal substitudes, namely those whose


unary operations are precisely those coming from C. These can also be characterised asthose for which the square constituting the unique substitude morphism to the terminalsubstitude 1 Ñ Comm is a pullback:

C //

��

P

��

1 // Comm.

The corresponding sub-2-category in RPat is the 2-full sub-2-category spanned bythe regular patterns SC Ñ M which are pullbacks of the terminal regular pattern S1 ÑFinSet. In detail, for any (strictified) regular pattern, which in virtue of Theorem 5.14is of the form SC Ñ FP , we have a commutative square of symmetric strong monoidalfunctors

SC //

��

FP

��

S1 // FinSet,

which clearly is a pullback precisely when the previous square is.The normality condition plays an important role in the theory of generalised mul-

ticategories of Cruttwell and Shulman [12], a more general framework which includessubstitudes as a special case. Their generalised multicategories are monoids in T -spans,for T a monad on a virtual equipment (a particular kind of double category). Theyare called normalised when the unary operations coincide with the vertical arrows in thedouble category, and the importance of the condition is that non-normalised generalisedmulticategories can be reinterpreted as normalised ones, by adjusting the monad andthe ambient virtual equipment. Cruttwell and Shulman also consider the discrete-objectscase, but do not consider the intermediate invertible case, which in the present situationis the most interesting.

5.18. Algebras for substitudes. Given a substitude φ : C Ñ P , and a symmetricmonoidal category W an algebra consists of

a functor A : C Ñ W for each x � px1, . . . , xnq and y P C, an action map

P px; yq ÝÑ W pApx1q b � � � b Apxnq, Apyqq

satisfying some conditions. One of these conditions says that for every arrow f : xÑ y inC, the action of the unary operation φpfq is equal to Apfq : Apxq Ñ Apyq. The remainingconditions are those of an algebra for the operad P ; in fact, the first condition implies thatfunctoriality in C follows from the P -algebra axioms, so to give an algebra for φ : C Ñ Pin W amounts just to give an algebra for the operad P in W .

Just as in the case of operads, the notion of algebra can also be described in terms of asubstitude of endomorphisms: the endomorphism substitude EA of a functor A : C Ñ W


is given in elementary terms as the substitude with category C and EApx1, . . . , xn; yq �W pApx1qb � � �bApxnq, Apyqq. An algebra structure on A : C Ñ W is now the same thingas an identity-on-C substitude map φ Ñ EA, that is, an operad map P Ñ EA under C.The endomorphism substitude can be described more conceptually as

C Ñ EndpW q|C,

so altogether the notion of algebra is conveniently formulated in terms of the substitudeadjunction: the endomorphism substitude of A : C Ñ W is precisely Endioppτ

Aq whereτA : SC Ñ W is the tautological factorisation of A through SC, and an algebra is anidentity-on-C substitude map φ Ñ Endioppτ

Aq. By adjunction, this is the same thing asan identity-on-C pinned symmetric strong monoidal functor

Fioppφq Ñ τA.

Giving this amounts just to giving α : FP Ñ W (a symmetric strong monoidal functor),that is, an algebra for the regular pattern corresponding to φ.

This works also for algebra homomorphisms: a homomorphism of substitude algebrasis a natural transformation u : A ñ B compatible with the action maps. In terms ofendomorphism substitudes, this compatibility amounts to commutativity of the diagram(of operads under C)

EA

up�,�q%%

P

;;

##

EA,B

EB up�,�q

99(13)

where EA,B is the W -bimodule SCop � C Ñ W given by EA,Bpx; yq � W pApx1q b � � � bApxnq, Bpyqq, up�, �q is induced by postcomposition with uy, and up�,�q is induced byprecomposition with ux1 b � � � b uxn . Now the natural transformation u : Añ B extendstautologically to a monoidal natural transformation τu : τA ñ τB, which in turn extendsto a monoidal natural transformation

FPα))

β

55ó W,

which is the corresponding homomorphism of regular-pattern algebras. The componentsof this monoidal natural transformation are the same as those of τu, since SC and FP havethe same object set; naturality in arrows in FP is a consequence of (13), and monoidalityfollows from construction.

Clearly this construction can be reversed, to construct an substitude-algebra homo-morphism from a monoidal natural transformation. Altogether we obtain the followingproposition.


5.19. Proposition. Algebras for a substitude are the same thing as algebras for thecorresponding regular pattern. More precisely for each symmetric monoidal category W ,there is an equivalence of categories between the category of algebras in W for a substitude,and the category of algebras in W for the corresponding regular pattern.

Since algebras for a substitude φ : C Ñ P are just algebras for the operad P , weimmediately get also, via the embedding Opd Ñ Subst by canonical groupoid pinning:

5.20. Corollary. Algebras for an operad are the same thing as algebras for the corre-sponding Feynman category.

Appendix A: Power coherence

In this appendix we recall coherence for symmetric monoidal categories, from the point ofview of Power’s general approach to coherence [33]. This point of view takes as input the 2-monad S on Cat for symmetric monoidal categories, and produces the coherence theoremfor symmetric monoidal categories. The formulation of this result given in Lemma A.4below, is most convenient for us for the purposes of studying regular patterns and Feynmancategories.

A.1. 2-monads and their algebras. A 2-monad T on a 2-category K is just a monadin the sense of Cat-enriched category theory. Thus one has the usual data of a monad

T : K ÝÑ K η : 1K ÝÑ T µ : T 2 ÝÑ T

but where T is a 2-functor, and η and µ are 2-natural, and the axioms are written downexactly as before. The extra feature is that now one has several different types of algebras,and several different types of algebra morphisms, and thus a variety of alternative 2-categories of algebras. For instance a pseudo T -algebra structure on A P K consists of thedata of an action a : TA Ñ A, as well as invertible coherence 2-cells a0 : 1A Ñ aηA anda2 : aT paq Ñ aµA, which satisfy:

a aηAa

a

a0a //

a2ηTA��''

id

�aT paqT 2paq aµAT

2paq

aµAµTAaT paqT pµAq

a2T 2paq//

a2µTA��

//

a2T pµAq

��aT pa2q �

aaT paqT pηAq

a

aT pa0qoo

a2T pηAq�� ww

id

�

When these coherence isomorphisms are identities, we have a strict T -algebra on A. Thereare also algebra types in which the coherence 2-cells are non-invertible, but these are notimportant for us here.

A lax morphism pA, aq Ñ pB, bq between pseudo T -algebras is a pair pf, fq, where


f : AÑ B and f : bT pfq Ñ fa, satisfying the following axioms:

f

bT pfqηA faηA

b0f

��

fηA

//��

fa0

�

bT pbqT 2pfq bµBT2pfq

faµA

faT paq

bT pfaq

b2T 2pfq//

fµA��

<<

fa2""fT paq

��bT pfq

�

When f is an isomorphism, f is said to be a pseudo morphism, and when f is an identity,f is said to be strict. Given lax T -algebra morphisms f and g : pA, aq Ñ pB, bq, a T -algebra 2-cell f Ñ g is a 2-cell φ : f Ñ g in K such that gpbT pφqq � pφaqf . In the caseT � S one has

lax morphism � symmetric lax monoidal functor, pseudo morphism � symmetric strong monoidal functor, strict morphism � symmetric strict monoidal functor, and algebra 2-cell � monoidal natural transformation.

The standard notations for some of the various 2-categories of algebras of a 2-monad Tare

Ps-T -Alg: objects are pseudo T -algebras, morphisms are pseudo morphisms, T -Algs: objects and morphisms are strict, and T -Algl: objects are strict and morphisms are lax,

A.2. The free symmetric-monoidal-category 2-monad. The 2-monad S is de-scribed explicitly in Section 5.1 of [36]. For a category C, an object of SC is a finitesequence of objects of C, and a morphism is of the form

pρ, pfiq1¤i¤nq : pxiq1¤i¤n ÝÑ pyiq1¤i¤n

where ρ P Σn is a permutation, and for i P n � t1, ..., nu, fi : xi Ñ yρi. Intuitively such amorphism is a permutation labelled by the arrows of C, as in

x1 x2 x3 x4

y1 y2 y3 y4.

((f1

��f2 �� f3

zzf4

The unit ηC : C Ñ SC is given by the inclusion of sequences of objects of length 1.The multiplication µC : S2C Ñ SC is given on objects by concatenation, and on arrowsby the substitution of labelled permutations. Given a symmetric monoidal category M ,pXiqi ÞÑ

ÂiXi is the effect on objects of a functor

Â: SM Ñ M , whose arrow map is


described using the symmetries of M . This is the action of a pseudo S-algebra structureon M . The unit coherences (those denoted a0 in the general definition) in this case areidentities, and the components of a2 come from the associators and unit coherences ofM . The strictness of M as a symmetric monoidal category, is the same thing as itsstrictness as an S-algebra. In fact, aside from the strictness of the unit coherences, andthe specification of the tensor product as an n-ary tensor product for all n, rather thanjust the cases n � 0 (the unit usually written as I) and n � 2 (the usual binary tensorproduct pA,Bq ÞÑ A b B), a pseudo S-algebra is exactly a symmetric monoidal categoryin the usual sense.

The coherence theorem for symmetric monoidal categories says that every symmetricmonoidal category is equivalent to a strict one. Mac Lane’s original proof of this involveda detailed combinatorial analysis. However from the point of view of Power’s generalapproach [33], this result comes out of how S interacts with the Gabriel factorisationsystem on Cat, by which every functor f : AÑ B is factored as

A C Bg// h //

where g is bijective on objects, and h is fully faithful.

A.3. The Gabriel factorisation system. The Gabriel factorisation system, in whichthe left class is that of bijective-on-objects functors, and the right class consists of thefully faithful functors, is an orthogonal factorisation system, meaning that arrows in theleft class admit a unique lifting property with respect to arrows in the right class. Oneway to formulate this, is to say that for any bijective-on-objects functor b : A Ñ B, andfully faithful functor f : C Ñ D, the square

CatpB,Cq CatpA,Cq

CatpA,DqCatpB,Dq

Catpb,Cq//

CatpA,fq��

//

Catpb,Dq

��CatpB,fq

is a pullback in the category of sets. In fact the Gabriel factorisation is enriched overCat, meaning that these squares are also pullbacks in Cat. Explicitly on arrows thissays that given α and β as in

A C

DB

))

f��))

��b

55

55

~�α

~� β

11??

��δ

such that fα � βb, there exists a unique δ as shown such that α � δb and fδ � β.Moreover the Gabriel factorisation system is an enhanced factorisation system in the


sense of Kelly [28], which is to say that given u, v and α as on the left,

A C

DB

u //

f��//

v

��b �

αA C

DB

u //

f��//

v

��b d

77

�β

there exists unique d and β as shown on the right, such that u � db and α � βb. It is notdifficult to verify that the Gabriel factorisation system enjoys these properties [34].

From the explicit description of S it is easy to see that S preserves bijective-on-objectsfunctors.

With these details in hand Power’s idea boils down to the following. Given a symmetricstrong monoidal functor F : S Ñ M where S is a symmetric strict monoidal category,one can take the Gabriel factorisation

S M 1 MG // H //

of F , and then factor the coherence witnessing F as strong monoidal on the left

SS S

MSM

Â//

F��//Â

��

SpF q �F �

SS S

M 1

MSM

SM 1

Â//

G��

H��//Â

��

SpHq

��

SpGq Â//

�

�H

uniquely as on the right, using enhancedness and the fact that SG is bijective on objects.

A.4. Lemma. [33] In the situation just described,Â

: SM 1 Ñ M 1 is a symmetric strictmonoidal structure on M 1. With respect to this structure, G is a symmetric strict monoidalfunctor, and H is the coherence datum making H into a symmetric strong monoidalfunctor.

We attribute this result to Power, although it is not exactly formulated in this way in[33]. Power’s result applies to more general monads T than S (only required to preservedbijective-on-objects functors), but for the functor F he only considers the case whereS � TM and F is the action TM ÑM . It is easy to adjust his argument to the presentsituation, to verify the strict S-algebra axioms for

Â: SM 1 Ñ M 1, and the pseudo

S-morphism axioms for H. Note that G is a strict S-algebra morphism by construction.

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Department of Mathematics, Macquarie University

Departament de matematiques, Universitat Autonoma de Barcelona

Faculty of Mathematics and Physics, Charles University, Prague

Email: [email protected]@mat.uab.cat

[email protected]

This article may be accessed at http://www.tac.mta.ca/tac/

https://hal.archives-ouvertes.fr/hal-00339331

https://hal.archives-ouvertes.fr/hal-00339331

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